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Search: id:A141712
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| A141712 |
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Triangle T, read by rows, where the n-th diagonal of T equals the BINOMIAL transform of the (n-1)-th diagonal of T^2 for n>=1, with the zeroeth diagonal set to all 1's, and where T^2 denotes the matrix square of T. |
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5
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| 1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 26, 26, 18, 8, 1, 162, 162, 114, 54, 16, 1, 1454, 1454, 1030, 506, 162, 32, 1, 18854, 18854, 13394, 6666, 2274, 486, 64, 1, 354258, 354258, 251962, 126134, 43798, 10346, 1458, 128, 1, 9671546, 9671546, 6882102, 3453110, 1210226
(list; table; graph; listen)
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OFFSET
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0,4
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FORMULA
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T(n,k) = Sum_{j=0..k} C(k,j)*[T^2](n-k+j-1,j) for n>k>=0, with T(n,n)=1, for n>=0.
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EXAMPLE
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Triangle T begins:
1;
1, 1;
2, 2, 1;
6, 6, 4, 1;
26, 26, 18, 8, 1;
162, 162, 114, 54, 16, 1;
1454, 1454, 1030, 506, 162, 32, 1;
18854, 18854, 13394, 6666, 2274, 486, 64, 1;
354258, 354258, 251962, 126134, 43798, 10346, 1458, 128, 1; ...
Matrix square T^2 begins:
1;
2, 1;
6, 4, 1;
26, 20, 8, 1;
162, 136, 68, 16, 1;
1454, 1292, 732, 236, 32, 1;
18854, 17400, 10648, 4036, 836, 64, 1; ...
where the BINOMIAL transform of diagonal 2 of T^2:
BINOMIAL[6,20,68,236,836,3020,11108,41516,...]
equals: [6,26,114,506,2274,10346,47634,221786,...]
which is diagonal 3 of T.
Specific examples:
T(4,1) = [T^2](2,0) + [T^2](3,1) = 6 + 20 = 26;
T(4,2) = [T^2](1,0) + 2*[T^2](2,1) + [T^2](3,2) = 2 + 2*4 + 8 = 18;
T(5,2) = [T^2](2,0) + 2*[T^2](3,1) + [T^2](4,2) = 6 + 2*20 + 68 = 114;
T(5,3) = [T^2](1,0) + 3*[T^2](2,1) + 3*[T^2](3,2) + [T^2](4,3) = 2 + 3*4 + 3*8 + 16 = 54.
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PROGRAM
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(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==k, 1, if(n==k+1, 2^k, if(k==1, T(n, 0), sum(j=0, k, binomial(k, j)*sum(i=0, n-k+j-1, T(n-k+j-1, i)*T(i, j)))))))}
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CROSSREFS
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Cf. A141713 (column 0), A141714 (column 2); A141715 (T^2), A141716.
Sequence in context: A121284 A108076 A104557 this_sequence A098539 A135880 A077873
Adjacent sequences: A141709 A141710 A141711 this_sequence A141713 A141714 A141715
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jul 01 2008
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